Is it obligatory for me to pray in the aforementioned mosque (1300 meters away, unofficial, a pre-built house, unofficial imam, no muezzin, and sometimes I find no one to pray with, especially at Fajr), or is it permissible for me to pray in congregation with my family at home?

A capable man must perform obligatory prayers in congregation at any mosque from which he hears the call to prayer. He is not permitted to pray at home except for a valid excuse. It is not a condition for the mosque to be official or to have a permanent imam; the criterion is the distance at which attending the congregation becomes difficult. If there is no severe hardship in going to the mosque, then prayer must be performed there, and if there is no muezzin, then you should make the call to prayer. However, if no one attends the prayer at the mosque, such as the dawn prayer, or if there is severe hardship, then there is no objection to praying in congregation at home.